A standard solution to general sceptical problems is to move to an externalist account of content. Grossly oversimplifying, if what makes a thought be about horses is that it has a causal connection with horses, then thoughts about horses can't be completely mistaken. This sort of move might be thought to be anti-realist, though I think that's a poor characterization. If this sort of move works, then we couldn't have thoughts and yet have our whole system of thoughts be completely mistaken. And hence, it seems, scepticism is dead.
But it just occurred to me that there is a hole in this argument. Why couldn't the sceptic who accepts the externalist story about content still say: "So, if I am thinking at all, then global scepticism is false. But am I thinking at all?" This may seem to be a completely absurd position—how could one doubt whether one is thinking? Wouldn't the doubt be a thought? Yes, the doubt would be a thought. Hence, the person who doubts whether she thinks would not be able to believe that she doubts. And, of course, the person who thinks she's not thinking has a contradiction between the content of her thought and the fact of her thought, but it's not so obvious that that's a contradiction in her thought (just as a contradiction between the content of an astronomical belief and an astronomical fact need not be a contradiction in the thinker's thought). Besides, the Churchlands think that they have no thoughts, and have given arguments for this.
If I am right in the above, then the content externalist move does not solve the problem of scepticism—it simply radicalizes it. But it raises the cost of scepticism—it forces the sceptic to stop thinking of herself as thinking. And as such it may be practically useful for curing scepticism if the sceptic isn't a full Pyrrhonian, in the way a rose or some other creature that has no thoughts is. However, if the motivation for the content externalism is to solve the problem of scepticism, rather than cure the sceptic, then the motivation seems to fail. (One difference between solving and curing is this. If a theory T solves a problem, then we have some reason to think T is true by inference to best explanation. But if believing a theory T would cure someone of a problem, inference to best explanation to the truth of T is not available. Though, still, I think the fact that believing T is beneficial would be some evidence for the truth of T in a world created by the good God.)
But it just occurred to me that there is a hole in this argument. Why couldn't the sceptic who accepts the externalist story about content still say: "So, if I am thinking at all, then global scepticism is false. But am I thinking at all?" This may seem to be a completely absurd position—how could one doubt whether one is thinking? Wouldn't the doubt be a thought? Yes, the doubt would be a thought. Hence, the person who doubts whether she thinks would not be able to believe that she doubts. And, of course, the person who thinks she's not thinking has a contradiction between the content of her thought and the fact of her thought, but it's not so obvious that that's a contradiction in her thought (just as a contradiction between the content of an astronomical belief and an astronomical fact need not be a contradiction in the thinker's thought). Besides, the Churchlands think that they have no thoughts, and have given arguments for this.
If I am right in the above, then the content externalist move does not solve the problem of scepticism—it simply radicalizes it. But it raises the cost of scepticism—it forces the sceptic to stop thinking of herself as thinking. And as such it may be practically useful for curing scepticism if the sceptic isn't a full Pyrrhonian, in the way a rose or some other creature that has no thoughts is. However, if the motivation for the content externalism is to solve the problem of scepticism, rather than cure the sceptic, then the motivation seems to fail. (One difference between solving and curing is this. If a theory T solves a problem, then we have some reason to think T is true by inference to best explanation. But if believing a theory T would cure someone of a problem, inference to best explanation to the truth of T is not available. Though, still, I think the fact that believing T is beneficial would be some evidence for the truth of T in a world created by the good God.)
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If this sort of move works, then we couldn't have thoughts and yet have our whole system of thoughts be completely mistaken. And hence, it seems, scepticism is dead.
If the whole system were mistaken, then externalism would be mistaken too. That's what the skeptic is having you entertain the possibility of. So the appeal to externalism doesn't seem to avoid the skeptical challenge.
But assume externalism is necessarily true. How does the possiblity that we are BIV's entail the possibility that we hae no thoughts? The possibility that we are BIV's entails that possibly part of the content of some thoughts is absent. We'd have incomplete content (sometimes), but we'd have thoughts. Some thoughts with incomplete content and some with complete content. One trivial case is the thought that 'I'm thinking'. Nothing incomplete there. A thought with incomplete content might include something like 'Bob's thinking'. Compare thoughts 'about' fictional beings, such as 'Anna K. killed herself'. I don't see how the latter two thoughts do not count as having thoughts, though I do see how it might undermine the intentionality of these thoughts.
The externalist could say: our concept of thought depends on a prior concept of a public language (cf. private language argument, Sellars, Brandom, etc.); if there are no public languages, we have no concept of them, and hence we have no concept of thought, either.
I think 'thought' like 'meaningful language' is more or less ambiguous. There is one sense in which something is a thought iff there are well-recognized inferential pathways to and from it. That is a more or less syntactic criterion. There is another sense in which something is a thought iff it has some connection with reality, with what it is putatively about. That is a more or less semantic criterion. Liar paradoxes are cases where these come apart--we have the syntactic inferential apparatus, but no semantic referential apparatus.
The content externalist needs the second notion: if one of our thoughts is about reality, then we can't be wholesale wrong. But what's available to introspection is (arguably, at most) the first notion: this thought has such-and-such as a consequence and is a consequence of so-and-so.
So I think you are right, both that the content externalist has raised the stakes for skepticism, and also that the content externalist doesn't get his premise for free.
It may, however, be essential to the inferential pathways being inferential that they are pathways between contentful states or that they have such-and-such causal connections with the external world. Take something like Brandom's inferentialism, but remove everything but one "mind". In particular, remove all outputs, and remove the causality in all inputs. There are still events that happen in the "mind" that are in some physicalist sense just like events that happen in our minds, and that correspond to noninferential belief formation in us of the sort that goes on in perception, but the noninferential beliefs now have no causes at all. (We're considering a radical sceptical hypothesis, after all, so we're free to suppose that our alleged perceptual inputs are just causeless brute events in our "minds".) I am not sure that inferentialism would survive this supposition. It seems to me that the notion of inputs and outputs, or of content, is going to be needed for an account of inferential practices.
I agree with you that it isn't "inference" properly speaking if the states are not contentful, and they wouldn't be in the case you describe. But I think that just goes to show that "inference" is itself an externalist property. There can be "narrow inferences" if you like, which would hold in the case you describe, which are (pretty much) purely syntactic and available to introspection.
I don't know about the "open to introspection". That's a tough question: would there be any consciousness at all in such a case?
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